The Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation — Paper

Abstract

The undecidability of a program's static semantic properties is governed by Rice's theorem. Self-modifying systems, however, require analysing not whether a property holds now, but whether it is preserved when the system rewrites itself. We formalise this transition through a semantic elevation operator {\Lambda}{\Phi}, which turns the static question "does x satisfy P?" into the dynamic question "is P preserved after x is transformed by {\Phi}?". We prove that when {\Phi} is intensional (depending on the source code, not only on the computed function), the elevated property remains undecidable even though it breaks the extensionality that Rice's theorem requires; the proof rests on Kleene's recursion theorem, not on Rice. Consequently the class U of non-verifiable properties is closed under the elevation operator. Unbounded iteration of the operator climbs the arithmetical hierarchy -to {\Pi}02-completeness- consolidating non-verifiability as a structural fact. We further show that the supervisory regress does not terminate: no fnite tower of increasingly capable verifiers yields an unconditional certificate. A categorical reading of these results in the efective topos, in which elevation appears as an instance of Lawvere's fxed-point theorem, is left as a direction for future work.

Authors 1

Jose Pascual Gumbau Mezquita

Specification

Official page: arxiv.org/abs/2609.11326
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Arxiv announce type: cross
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
arXiv id: 2609.11326
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Categories: cs.LO, cs.AI, cs.CL, math.LO
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
PDF: arxiv.org/pdf/2609.11326
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Primary category: cs.LO
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Published: 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh

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9

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T19

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6 h ago

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Authors: Jose Pascual Gumbau Mezquita

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Value	Valid from → to	Status	Source	Confidence	Extractor
https://arxiv.org/abs/2609.11326	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Claims are temporal and append-only: a new observation closes the previous claim (valid_to) instead of overwriting it. Conflicting claims from different sources are kept side by side and flagged — never averaged. Methodology →

11 Sept 2026
New paperPaperThe Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation
New paper: The Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation
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arXiv (Atom API + RSS)	rss.arxiv.org/rss/cs.CL	feed	T1· Official	4 h ago	1

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